Simplify (x^2+5x+4)/(x^2-8x)*(6x^2-102x+432)/(4x^2-20x-144)

$\frac{x^{2} + 5 x + 4}{x^{2} - 8 x} \cdot \frac{6 x^{2} - 102 x + 432}{4 x^{2} - 20 x - 144}$

Step 1

Factor $x^{2} + 5 x + 4$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $4$ and whose sum is $5$ .

$1, 4$

Write the factored form using these integers.

$\frac{(x + 1) (x + 4)}{x^{2} - 8 x} \cdot \frac{6 x^{2} - 102 x + 432}{4 x^{2} - 20 x - 144}$

Step 2

Factor $x$ out of $x^{2} - 8 x$ .

Factor $x$ out of $x^{2}$ .

$\frac{(x + 1) (x + 4)}{x \cdot x - 8 x} \cdot \frac{6 x^{2} - 102 x + 432}{4 x^{2} - 20 x - 144}$

Factor $x$ out of $- 8 x$ .

$\frac{(x + 1) (x + 4)}{x \cdot x + x \cdot - 8} \cdot \frac{6 x^{2} - 102 x + 432}{4 x^{2} - 20 x - 144}$

Factor $x$ out of $x \cdot x + x \cdot - 8$ .

$\frac{(x + 1) (x + 4)}{x (x - 8)} \cdot \frac{6 x^{2} - 102 x + 432}{4 x^{2} - 20 x - 144}$

Step 3

Simplify the numerator.

Factor $6$ out of $6 x^{2} - 102 x + 432$ .

Factor $6$ out of $6 x^{2}$ .

$\frac{(x + 1) (x + 4)}{x (x - 8)} \cdot \frac{6 (x^{2}) - 102 x + 432}{4 x^{2} - 20 x - 144}$

Factor $6$ out of $- 102 x$ .

$\frac{(x + 1) (x + 4)}{x (x - 8)} \cdot \frac{6 (x^{2}) + 6 (- 17 x) + 432}{4 x^{2} - 20 x - 144}$

Factor $6$ out of $432$ .

$\frac{(x + 1) (x + 4)}{x (x - 8)} \cdot \frac{6 x^{2} + 6 (- 17 x) + 6 \cdot 72}{4 x^{2} - 20 x - 144}$

Factor $6$ out of $6 x^{2} + 6 (- 17 x)$ .

$\frac{(x + 1) (x + 4)}{x (x - 8)} \cdot \frac{6 (x^{2} - 17 x) + 6 \cdot 72}{4 x^{2} - 20 x - 144}$

Factor $6$ out of $6 (x^{2} - 17 x) + 6 \cdot 72$ .

$\frac{(x + 1) (x + 4)}{x (x - 8)} \cdot \frac{6 (x^{2} - 17 x + 72)}{4 x^{2} - 20 x - 144}$

Factor $x^{2} - 17 x + 72$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $72$ and whose sum is $- 17$ .

$- 9, - 8$

Write the factored form using these integers.

$\frac{(x + 1) (x + 4)}{x (x - 8)} \cdot \frac{6 ((x - 9) (x - 8))}{4 x^{2} - 20 x - 144}$

$\frac{(x + 1) (x + 4)}{x (x - 8)} \cdot \frac{6 (x - 9) (x - 8)}{4 x^{2} - 20 x - 144}$

Step 4

Simplify the denominator.

Factor $4$ out of $4 x^{2} - 20 x - 144$ .

Factor $4$ out of $4 x^{2}$ .

$\frac{(x + 1) (x + 4)}{x (x - 8)} \cdot \frac{6 (x - 9) (x - 8)}{4 (x^{2}) - 20 x - 144}$

Factor $4$ out of $- 20 x$ .

$\frac{(x + 1) (x + 4)}{x (x - 8)} \cdot \frac{6 (x - 9) (x - 8)}{4 (x^{2}) + 4 (- 5 x) - 144}$

Factor $4$ out of $- 144$ .

$\frac{(x + 1) (x + 4)}{x (x - 8)} \cdot \frac{6 (x - 9) (x - 8)}{4 x^{2} + 4 (- 5 x) + 4 \cdot - 36}$

Factor $4$ out of $4 x^{2} + 4 (- 5 x)$ .

$\frac{(x + 1) (x + 4)}{x (x - 8)} \cdot \frac{6 (x - 9) (x - 8)}{4 (x^{2} - 5 x) + 4 \cdot - 36}$

Factor $4$ out of $4 (x^{2} - 5 x) + 4 \cdot - 36$ .

$\frac{(x + 1) (x + 4)}{x (x - 8)} \cdot \frac{6 (x - 9) (x - 8)}{4 (x^{2} - 5 x - 36)}$

Factor $x^{2} - 5 x - 36$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 36$ and whose sum is $- 5$ .

$- 9, 4$

Write the factored form using these integers.

$\frac{(x + 1) (x + 4)}{x (x - 8)} \cdot \frac{6 (x - 9) (x - 8)}{4 ((x - 9) (x + 4))}$

$\frac{(x + 1) (x + 4)}{x (x - 8)} \cdot \frac{6 (x - 9) (x - 8)}{4 (x - 9) (x + 4)}$

Step 5

Simplify terms.

Cancel the common factor of $x + 4$ .

Factor $x + 4$ out of $(x + 1) (x + 4)$ .

$\frac{(x + 4) (x + 1)}{x (x - 8)} \cdot \frac{6 (x - 9) (x - 8)}{4 (x - 9) (x + 4)}$

Factor $x + 4$ out of $4 (x - 9) (x + 4)$ .

$\frac{(x + 4) (x + 1)}{x (x - 8)} \cdot \frac{6 (x - 9) (x - 8)}{(x + 4) (4 (x - 9))}$

Cancel the common factor.

$\frac{(x + 4) (x + 1)}{x (x - 8)} \cdot \frac{6 (x - 9) (x - 8)}{(x + 4) (4 (x - 9))}$

Rewrite the expression.

$\frac{x + 1}{x (x - 8)} \cdot \frac{6 (x - 9) (x - 8)}{4 (x - 9)}$

Cancel the common factor of $x - 8$ .

Factor $x - 8$ out of $x (x - 8)$ .

$\frac{x + 1}{(x - 8) x} \cdot \frac{6 (x - 9) (x - 8)}{4 (x - 9)}$

Factor $x - 8$ out of $6 (x - 9) (x - 8)$ .

$\frac{x + 1}{(x - 8) x} \cdot \frac{(x - 8) (6 (x - 9))}{4 (x - 9)}$

Cancel the common factor.

$\frac{x + 1}{(x - 8) x} \cdot \frac{(x - 8) (6 (x - 9))}{4 (x - 9)}$

Rewrite the expression.

$\frac{x + 1}{x} \cdot \frac{6 (x - 9)}{4 (x - 9)}$

Multiply $\frac{x + 1}{x}$ by $\frac{6 (x - 9)}{4 (x - 9)}$ .

$\frac{(x + 1) (6 (x - 9))}{x (4 (x - 9))}$

Cancel the common factor of $6$ and $4$ .

Factor $2$ out of $(x + 1) (6 (x - 9))$ .

$\frac{2 ((x + 1) (3 (x - 9)))}{x (4 (x - 9))}$

Cancel the common factors.

Factor $2$ out of $x (4 (x - 9))$ .

$\frac{2 ((x + 1) (3 (x - 9)))}{2 (x (2 (x - 9)))}$

Cancel the common factor.

$\frac{2 ((x + 1) (3 (x - 9)))}{2 (x (2 (x - 9)))}$

Rewrite the expression.

$\frac{(x + 1) (3 (x - 9))}{x (2 (x - 9))}$

Cancel the common factor of $x - 9$ .

Cancel the common factor.

$\frac{(x + 1) (3 (x - 9))}{x (2 (x - 9))}$

Rewrite the expression.

$\frac{(x + 1) \cdot (3)}{x \cdot (2)}$

Reorder.

Move $3$ to the left of $x + 1$ .

$\frac{3 \cdot (x + 1)}{x \cdot (2)}$

Move $2$ to the left of $x$ .

$\frac{3 (x + 1)}{2 x}$

Simplify (x^2+5x+4)/(x^2-8x)*(6x^2-102x+432)/(4x^2-20x-144)

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